Solution Of Advection-diffusion Equation Research Articles

Analytical solution of advection diffusion equation in two dimensions using different shapes of wind speed and eddy diffusivity

The diffusion equation has been derived in two dimensions using two methods: variable separation and substituting. We compared the results from these solutions to the results from the Copenhagen experiment, taking into consideration the differences in wind speed and eddy diffusivity.

A Comparison between Gaussian Plume Models and Analytical Advection-Diffusion Equation in Unstable Condition

In this paper, comparing between Gaussian plume models and the solution of Advection-diffusion equation in three dimensions using different shapes of dispersion parameters and eddy diffusivities respectively. After that, comparing between Gaussian plume model, proposed model and observed concentrations data which measuring on Egyptian Atomic Energy Authority EAEA for Iodine-135 (I135) in unstable condition.

Remediation of pollution in a river by releasing clean water using the solution of advection-diffusion equation in two dimensions

Remediation of pollution in a river by releasing clean water using the solution of advection-diffusion equation in two dimensions

Numerical solutions of advection diffusion equations involving Atangana–Baleanu time fractional derivative via cubic B-spline approximations

The B-spline or the basis spline function, is a piecewise polynomial function made up of polynomials, its domain is subdivided by knots, and basis functions are non-zero on the entire domain. In this study, a new cubic B-spline (NCBS) approximation together with the θ-weighted scheme is formed to approximate the numerical solution of the time fractional advection diffusion equation (TFADE) involving the Atangana–Baleanu time fractional derivative (ABTFD). The finite difference scheme (FDS) is utilized to discretize the ABTFD with a non-singular kernel. The spatial derivative is discretized by using NCBS functions. The stability analysis of the proposed technique is studied. Convergence analysis of the current technique is also analyzed. The proposed technique is examined on a variety of problems, and the numerical outcomes are contrasted with the previously published technique’s results to ensure the correctness and accuracy of the current technique.

A brief review on the solutions of advection-diffusion equation

In this work both linear and nonlinear advection-diffusion equations are considered and discussed their analytical solutions with different initial and boundary conditions. The work of Ogata and Banks, Harleman and Rumer, Cleary and Adrian, Atul Kumar et al., Mojtabi and Deville are reviewed for linear advection-diffusion equations and for nonlinear, we have chosen the work of Sakai and Kimura. Some enthusiastic functions used in the articles, drawbacks and applications of the results are discussed. Reduction of the advection-diffusion equations into diffusion equations make the governing equation solvable by using integral transform method for analytical solution. For nonlinear advection-diffusion equations, the Cole-Hopf transformation is used to reduce into the diffusion equation. Different dispersion phenomena in atmosphere, surface and subsurface area are outlined.

Investigation of Water Pollution in the River with Second-Order Explicit Finite Difference Scheme of Advection-Diffusion Equation and First-Order Explicit Finite Difference Scheme of Advection-Diffusion Equation

The perseverance of this research article is to investigate water pollution in rivers with a second-order explicit finite difference scheme of advection-diffusion equation (ADE) and a first-order explicit finite difference scheme of ADE. For investigation, two numerical schemes exploit here FTCSCS and second-order Lax-Wendroff type of ADE which is our new proposed one. In earlier Lax-Wendroff, type scheme existed only for hyperbolic partial differential equation (PDE), here a new second-order Lax-Wendroff type scheme is proposed for parabolic PDE and in addition assist to investigate water pollution with an expectation of better yield compared to the existing one. We implement numerical schemes to estimate the pollutant in water at different times and different points of water bodies. We investigate the numerical behaviour of water pollution by implementing the explicit centred difference scheme (FTCSCS) for advection-diffusion and for our proposed second-order Lax-Wendroff type scheme. Our computational result verifies the qualitative behavior of the solution of ADE for various considerations of the parameters.

Evaluation of analytical solution of advection diffusion equation in three dimensions

AbstractThree‐dimensional advection–diffusion equation with steady state was evaluated from continuous point source taking the vertical and crosswind eddy diffusivities as power law of vertical height and downwind distance with constant wind speed to get the concentration in three dimensions. Separation of variables technique and Hankel transform were used to calculate this equation. The proposed analytical solution was compared with diffusion experiment of radioactive Iodine‐135 (I135) in unstable condition at Inshas, Cairo, Egypt. A good agreement achieved between calculated and observed concentration support our proposed model.

Solving fractional Advection-diffusion equation using Genocchi operational matrix based on Atangana-Baleanu derivative

<p style='text-indent:20px;'>In recent years, a new definition of fractional derivative which has a nonlocal and non-singular kernel has been proposed by Atangana and Baleanu. This new definition is called the Atangana-Baleanu derivative. In this paper, we present a new technique to obtain the numerical solution of advection-diffusion equation containing Atangana-Baleanu derivative. For this purpose, we use the operational matrix of fractional integral based on Genocchi polynomials. An error bound is given for the approximation of a bivariate function using Genocchi polynomials. Finally, some examples are given to illustrate the applicability and efficiency of the proposed method.</p>

Numerical solution of advection-diffusion equation using finite difference schemes

This paper presents numerical simulation of one-dimensional advection-diffusion equation. We study the analytical solution of advection diffusion equation as an initial value problem in infinite space and realize the qualitative behavior of the solution in terms of advection and diffusion co-efficient. We obtain the numerical solution of this equation by using explicit centered difference scheme and Crank-Nicolson scheme for prescribed initial and boundary data. We implement the numerical scheme by developing a computer programming code and present the stability analysis of Crank-Nicolson scheme for ADE. For the validity test, we perform error estimation of the numerical scheme and presented the numerical features of rate of convergence graphically. The qualitative behavior of the ADE for different choice of the advection and diffusion co-efficient is verified. Finally, we estimate the pollutant in a river at different times and different points by using these numerical scheme.
 Bangladesh J. Sci. Ind. Res.55(1), 15-22, 2020

Mathematical Models for Calculation of Mixing Zones for Coastal Effluent Discharges

The potential long-term environmental impacts of coastal effluent discharges can be addressed and regulated using a mixing zone concept to control and manage the mixing capacity of the receiving coastal waters. The standard regulatory guidelines consist of two key elements: a concentration limit and a point of compliance expressed as a radius centered at the end of the outfall pipe. Modeling studies of the effect of seabed depth upon dispersion of coastal effluent discharges into the sea in the far-field are investigated analytically using the solutions of two-dimensional advection-diffusion equations with a point source on the simple depth profiles of a flat seabed and a uniformly sloping seabed. Solutions are then illustrated graphically by plotting contours of concentration, showing that the effluent discharged plumes are spreading downstream and heading towards the shoreline. Based on the location of maximum value of concentration at the shoreline, the imposed radius of and limit of concentration level within the circular mixing zone are formulated to be used as a measure for assessing the impact of effluent discharges in the coastal environment. The results found agreed with the mixing zones implemented in practice as the environmental quality standards by the regulatory authorities to minimize the impact. As the sea outfall's long pipeline may go beyond the continental shelf, a sudden step change in the seabed depth profiles is introduced, where the solutions of the advection-diffusion equations are obtained using the method of images. An extended formulation of the mixing zones radius and concentration limit are also presented.

An ADMM-AHM Integrated Approach for Problems with Rapidly Oscillating Coefficients

The Advection-Diffusion Multilayer Method (ADMM) emerged to address the solution of advection-diffusion equations with variable coefficients in the context of pollutant dispersion modeling. The ADMM is based on the piecewise-constant approximation of the variable coefficients and the application of the Laplace transform. Applications of ADMM in other areas are potentially relevant for modeling the behavior of heterogeneous media. However, if such heterogeneity is characterized by rapidly oscillating coefficients, the direct application of the ADMM would increase the computational effort needed, as it would require a very fine discretization of the domain. In order to overcome such a drawback, in this contribution, an alternative approach combining the ADMM with the Asymptotic Homogenization Method (AHM) is presented. The ADMM-AHM integrated approach is compared to the direct application of the ADMM in order to assess the accuracy of the estimations of the solution of the original problem, and the computational efficiency.

An efficient Bi-cubic B-spline ADI method for numerical solutions of two-dimensional unsteady advection diffusion equations

PurposeThis paper aims to develop a novel numerical method based on bi-cubic B-spline functions and alternating direction (ADI) scheme to study numerical solutions of advection diffusion equation. The method captures important properties in the advection of fluids very efficiently. C.P.U. time has been shown to be very less as compared with other numerical schemes. Problems of great practical importance have been simulated through the proposed numerical scheme to test the efficiency and applicability of method.Design/methodology/approachA bi-cubic B-spline ADI method has been proposed to capture many complex properties in the advection of fluids.FindingsBi-cubic B-spline ADI technique to investigate numerical solutions of partial differential equations has been studied. Presented numerical procedure has been applied to important two-dimensional advection diffusion equations. Computed results are efficient and reliable, have been depicted by graphs and several contour forms and confirm the accuracy of the applied technique. Stability analysis has been performed by von Neumann method and the proposed method is shown to satisfy stability criteria unconditionally. In future, the authors aim to extend this study by applying more complex partial differential equations. Though the structure of the method seems to be little complex, the method has the advantage of using small processing time. Consequently, the method may be used to find solutions at higher time levels also.Originality/valueADI technique has never been applied with bi-cubic B-spline functions for numerical solutions of partial differential equations.

Analytical Solutions of ADE with Temporal Coefficients for Continuous Source in Infinite and Semi-Infinite Media

AbstractIn the present technical note, two aspects commonly used to obtain the analytical solution of advection–diffusion equation (ADE) with time-dependent coefficients describing solute transport...

The 5th International Conference on Science & Engineering in Mathematics, Chemistry and Physics 2017 (ScieTech 2017), The Ramada Bintang Hotel, Kuta, Bali, Indonesia 21 - 22 January 2017

The 5th International Conference on Science & Engineering in Mathematics, Chemistry and Physics 2017 (ScieTech 2017), The Ramada Bintang Hotel, Kuta, Bali, Indonesia 21 - 22 January 2017

Comparative Analysis of Numerical Solutions of Advection-Diffusion Equation

Abstract. The advection diffusion equation is one of the most popular and convenient equations in calculating the transport of energy and materials in flux areas. In this paper, one-dimensional advection-diffusion equation is solved using the finite difference, fourth order finite difference, finite volume, and differential quadrature methods in explicit condition. The quantitative comparative analysis involved two hypothetical cases and one experimental study. The results of the numerical solutions for the hypothetical cases are compared against the analytical solution. The experimental data are also simulated by the methods. The comparative analysis results revealed that the differential quadrature method performs as good as the analytical solution for the hypothetical cases. All the methods but the finite difference showed comparable performance in simulating the experimental data. Keywords: Advection-diffusion equation, Numerical methods, Explicit solution Ozet. Tasinim-yayilim denklemi, akim alanlarinda enerji ve malzemelerin tasiniminin hesaplanmasinda kullanilan en populer ve kullanisli denklemlerden biridir. Bu makalede, bir boyutlu tasinim-yayilim denkleminin, sonlu fark, dorduncu mertebeden sonlu farklar, sonlu hacim ve diferansiyel kuadratur yontemleri kullanilarak acik cozumleri yapilmistir. Kantitatif karsilastirmali inceleme, iki varsayimsal vaka ile bir deneysel calisma icermektedir. Varsayimsal vakalar icin sayisal cozumlerin sonuclari analitik cozum ile karsilastirilmistir. Deneysel veriler de yontemlerle simule edilmistir. Karsilastirmali inceleme sonuclari, diferansiyel kuadratur yonteminin varsayimsal durumlarda analitik cozum kadar iyi performans gosterdigini ortaya koymustur. Cozumde kullanilan tum yontemler deneysel verilerin simulasyonu icin karsilastirmaya deger performans gostermistir. Anahtar Sozcukler: Tasinim-yayilim denklemi, Sayisal yontemler, Acik cozum

Statistical Downscaling of Air Dispersion Model Using Neural Network for Delhi

ABSTRACTStatistical downscaling methods are used to extract high resolution information from coarse resolution models. The accuracy of a modelling system in analyzing the issues of either continuous or accidental release in the atmosphere is important especially when adverse health effects are expected to be found. Forecasting of air quality levels are commonly performed with either deterministic or statistical. In this study, statistical downscaling approach is investigated for hourly PM10 (particulate matter with aerodynamic diameter < 10 µm) pollutant for Delhi. The statistical downscaling is used on air dispersion model using neural network technique. The air dispersion model is based on analytical solution of advection diffusion equation in Neumann boundary condition for a bounded domain. Power laws are assumed for height dependent wind speed; and downwind and vertical eddy diffusivities are considered as an explicit function of downwind distance and vertical height. The predicted concentration of dispersion model with meteorological variables is used as input parameters to the neural network. It is found that performance of both air dispersion model and “pure” statistical models is inferior to that of the statistical downscaled model. In particular the root mean squares error (RMSE) of the deterministic model is reduced by at least 35% and 45% for hourly and rush hours particulate matter concentrations respectively using statistical downscaling. In addition, the results with statistical downscaled method show that the errors of the forecasts are reduced by at least 30% for stable and unstable-neutral atmospheric conditions.

Coupling continuum dislocation transport with crystal plasticity for application to shock loading conditions

We have developed a multi-physics modeling approach that couples continuum dislocation transport, nonlinear thermoelasticity, crystal plasticity, and consistent internal stress and deformation fields to simulate the single-crystal response of materials under extreme dynamic conditions. Dislocation transport is modeled by enforcing dislocation conservation at a slip-system level through the solution of advection-diffusion equations. Nonlinear thermoelasticity provides a thermodynamically consistent equation of state to relate stress (including pressure), temperature, energy densities, and dissipation. Crystal plasticity is coupled to dislocation transport via Orowan's expression where the constitutive description makes use of recent advances in dislocation velocity theories applicable under extreme loading conditions. The configuration of geometrically necessary dislocation density gives rise to an internal stress field that can either inhibit or accentuate the flow of dislocations. An internal strain field associated with the internal stress field contributes to the kinematic decomposition of the overall deformation. The paper describes each theoretical component of the framework, key aspects of the constitutive theory, and some details of a one-dimensional implementation. Results from single-crystal copper plate impact simulations are discussed in order to highlight the role of dislocation transport and pile-up in shock loading regimes. The main conclusions of the paper reinforce the utility of the modeling approach to shock problems.

Construction of locally conservative fluxes for the SUPG method

We consider the construction of locally conservative fluxes by means of a simple postprocessing technique obtained from the finite element solutions of advection diffusion equations. It is known that a naive calculation of fluxes from these solutions yields nonconservative fluxes. We consider two finite element methods: the usual continuous Galerkin finite element method for solving nondominating advection diffusion equations and the streamline upwind/Petrov‐Galerkin method for solving advection dominated problems. We then describe the postprocessing technique for constructing conservative fluxes from the numerical solutions of the general variational formulation. The postprocessing technique requires solving an auxiliary Neumann boundary value problem on each element independently and it produces a locally conservative flux on a vertex centered dual mesh relative to the finite element mesh. We provide a convergence analysis for the postprocessing technique. Performance of the technique and the convergence behavior are demonstrated through numerical examples including a set of test problems for advection diffusion equations, advection dominated equations, and drift‐diffusion equations. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1971–1994, 2015

Analysis of Navier-Stokes Equation from the Viewpoint of Advection Diffusion

We propose a highly accurate approximate solution for Navier-Stokes Equation (NSE) based on the similarity between NSE and Advection Diffusion Equation (ADE). First, we present the analytical exact solution of ADE using a Green function (integral kernel) that is obtained from the diffusion equation over uniform flow field (or velocity field) in three dimensional (3D) boundless region under arbitrary initial condition. Next, from the explicit similarity between NSE and ADE, we derive the approximate solution of NSE using the analytical solution of ADE.

Numerical simulation of advection-diffusion equation

This paper presents numerical solution of advection-diffusion equation (ADE) using B-spline finite element method. The one dimensional ADE describes many quantities such as vorticity, heat, mass, energy, velocity etc. Test example shows the accuracy of the numerical technique where different comparisons are made for the exact and approximate solution obtained. Numerical results are found to be in good agreement with the analytical results.